Properties

Modulus 450
Conductor 45
Order 12
Real no
Primitive no
Minimal yes
Parity even
Orbit label 450.p

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(450)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,9]))
 
pari: [g,chi] = znchar(Mod(443,450))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 450
Conductor = 45
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 450.p
Orbit index = 16

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{450}(257,\cdot)\) \(\chi_{450}(293,\cdot)\) \(\chi_{450}(407,\cdot)\) \(\chi_{450}(443,\cdot)\)

Values on generators

\((101,127)\) → \((e\left(\frac{1}{6}\right),-i)\)

Values

-117111317192329313741
\(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(-i\)\(e\left(\frac{5}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 450 }(443,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{450}(443,\cdot)) = \sum_{r\in \Z/450\Z} \chi_{450}(443,r) e\left(\frac{r}{225}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 450 }(443,·),\chi_{ 450 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{450}(443,\cdot),\chi_{450}(1,\cdot)) = \sum_{r\in \Z/450\Z} \chi_{450}(443,r) \chi_{450}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 450 }(443,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{450}(443,·)) = \sum_{r \in \Z/450\Z} \chi_{450}(443,r) e\left(\frac{1 r + 2 r^{-1}}{450}\right) = -0.0 \)